PSI - Issue 79

Abubakr E.S. Musa et al. / Procedia Structural Integrity 79 (2026) 206–216

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To model an equivalent imperfection that accurately represents all the initial imperfections observed in the test specimens, two key characteristics must be considered: shape and amplitude . Regarding the shape, it has been observed that using eigenmode imperfections distributes equivalent imperfections across the tube's geometry. However, this distribution could interact unfavorably with the intended thickness reduction to be applied later in the analysis. As a result, all imperfections distributed across the tube's surface are excluded as viable options. Moreover, tests conducted on specimens T1-1 and T1-2 revealed that the specimens exhibit an axisymmetric pattern. Based on this observation, the shape of the equivalent imperfections is selected to follow an axisymmetric configuration, specifically targeting the upper and lower regions of the tube. The selected shape consists of three half waves at the top and three half-waves at the bottom, with the length of each half-wave defined as 1.728 √ Rt (Rotter (2003)), where R is the radius, and t is the thickness of the tube. This approach ensures that the modeled imperfections closely align with the observed behavior of the test specimens while avoiding interference with the subsequent application of thickness reduction. To generate the imperfection with the specified shape, the nodes translation technique, recently developed by Musa et al. (2021), was employed. This method involves first creating a perfect shell model that incorporates the material properties and activates the geometric nonlinear analysis option. Subsequently, a computational platform, such as Wolfram Mathematica or MATLAB, is utilized to calculate the nodes coordinates for the imperfect shell based on the desired imperfection profile. The generated coordinates of the imperfect shell nodes are then used to replace the corresponding nodes coordinates in the input file (.INP) of the perfect shell model. This procedure transforms the input file of the perfect shell model into an input file of the imperfect shell model, accurately accounting for the equivalent geometric imperfection. Fig. 3 (a) illustrates the nodes generated using Mathematica to define the imperfection shape, while Fig. 3 (b) displays the modified model after the altered input file is imported into ABAQUS. It is important to note that these figures are magnified in the radial direction to enhance the visualization of the imperfection mode. For reference, Fig. 3 (c) also provides an example of the imperfection profile at its original scale, without magnification, to facilitate a more precise understanding of its geometry.

Fig. 3. Generation of the imperfect shell model with equivalent geometric imperfection